De Morgan Semi-Heyting and Heyting Algebras

TL;DR

This paper investigates the properties of De Morgan semi-Heyting and Heyting algebras, proving that certain varieties satisfy the Stone identity at specific levels, and establishing the level of these algebraic structures.

Contribution

It proves that DMSH1-algebras of level 1 satisfy Stone identity and that DmsStSH and De Morgan semi-Heyting algebras are at level 2, extending previous results.

Findings

DMSH1-algebras of level 1 satisfy Stone identity

DmsStSH variety is at level 2

De Morgan semi-Heyting algebras are at level 2

Abstract

The variety DMSH of semi-Heyting algebras with a De Morgan negation was introduced in [12] and an increasing sequence DMSHn of level n, n being a natural number, of its subvarieties was investigated in the series [12], [13], [14], [15], [16], and [17], of which the present paper is a sequel. In this paper, we prove two main results: Firstly, we prove that DMSH1-algebras of level 1 satisfy Stone identity, generalizing an earlier result that regular DMSH1-algebras of level 1 satisfy Stone identity. Secondly, we prove that the variety of DmsStSH of dually ms, Stone semi-Heyting algebras is at level 2. As an application, it is derived that the variety of De Morgan semi-Heyting algebras is also at level 2. It is also shown that these results are sharp.